On Some Properties of Functions on Convex Galaxies

Abstract

In this paper, we define and study extensively a new type of external sets in , we call it "convex galaxies". We show that these convex external sets may be classified in some definite types. More precisely, we obtain the following : (1)Let be a convex galaxy which is symmetric with respect to zero, then (i) is an - galaxy (0) if and only if there exists an internal strictly increasing sequence of strictly positive real numbers with such that and , for all , where, is some limited real number such that . (ii) is a non-linear galaxy if and only if there exists an internal strictly increasing sequence of strictly positive real numbers with such that is unlimited for all . (2)Let be a convex galaxy which is symmetric with respect to zero, then (i) is an - galaxy (0) iff there exists a real internal strictly increasing - function , such that , and for all limited , where is a positive real number. (ii) is a non-linear galaxy if and only if there exists a real internal strictly increasing - function , such that and is positive unlimited, for all appreciable .

Keywords

• convex
• galaxy
• external sets